MPs have renewed their call for the Speaker, John Bercow, to rethink the appointment of an Australian as the House of Commons clerk, parliament's most senior official, following the revelation that he has said he does not believe the job should be done by one person.

Bercow told the Tory MP Sir Alan Duncan in a letter sent on Thursday that the two parts of the job – running the House of Commons administratively and advising MPs on parliamentary procedure – should be separated "as soon as practical" and that he thinks this matter could be addressed "swiftly".

A committee overseen by Bercow chose Carol Mills, head of the department of parliamentary services in Canberra, to replace the outgoing clerk, Sir Robert Rogers. But the appointment provoked a cross-party campaign opposing her appointment. Some MPs said Mills lacked experience of overseeing parliamentary procedure and guarding MPs' privileges.

The appointment is yet to be confirmed by the Queen, a process triggered when No 10 sends the name of the successful candidate to Buckingham Palace. Some MPs are demanding a confirmation hearing in the hope this could be used to stop Mills.

Bernard Jenkin, the chairman of the Commons public administration committee, said Sunday's revelation about Bercow wanting the job to be split was adding "confusion" to the process and amounted to "a tacit admission that the appointment may be far from ideal".

He said that the decision should be blocked and it was "time to go back to the drawing board".

But Bercow believes that it is too late to halt the process and that MPs are criticising Mills without knowing how well she performed at interview. No 10 has no intention of holding up her appointment, according to a source close to the Speaker.

"[Those opposed to Mills] are all people who were outside the interview room, who did not get to see the applications submitted and how well people performed, not just in one interview, but in two. The panel who chose her were incredibly mindful that the role has two distinct elements," the source said.

Bercow believes that opposition to Mills is partly driven by MPs not being fully aware of how important the administrative aspect of the clerk's job is, the source said. "They just saw this person in a wig who sits at the table in the chamber and they had not cottoned on to the size of the job, the chief executive element."

Bercow believes that there is growing support in the Commons for the idea of splitting the job, but he does not favour a confirmation hearing. "It's a pity nobody suggested it before the process started," said the source. "It's a bit late for it this time round."

In his letter, the Speaker said: "It has disappointed me that when I previously floated the idea [of splitting the job] with colleagues – most recently with Andrew Lansley when leader of the House and with our retiring clerk, Sir Robert Rogers – it has not been met with much (or even any) enthusiasm."

• This article was amended on 26 August 2014. We initially said Bernard Jenkins is head of the public accounts committee